On Lagrangian–Grassmannian codes

Carrillo-Pacheco, Jesús; Zaldívar, Felipe

doi:10.1007/s10623-010-9434-4

Cited by 19 publications

(32 citation statements)

References 16 publications

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“…This is the Grassmannian image of the space of maximal totally isotropic (Lagrangian) subspaces of P G(2n − 1, 2), i.e. the image of the Lagrangian Grassmannian [64]. Since Lagrangian subspaces are (locally) represented by the arrangement (1|A) where A is symmetric, our real slice is an algebraic subvariety of the bulk of dimension n(n + 1)/2.…”

Section: The Real Slice Of the Bulkmentioning

confidence: 99%

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Finite geometric toy model of spacetime as an error correcting code

Lévay

Holweck

2019

Phys. Rev. D

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A finite geometric model of space-time (which we call the bulk) is shown to emerge as a set of error correcting codes. The bulk is encoding a set of messages located in a blow up of the Gibbons-Hoffman-Wootters (GHW) discrete phase space for n-qubits (which we call the boundary). Our error correcting code is a geometric subspace code known from network coding, and the correspondence map is the finite geometric analogue of the Plücker map well-known form twistor theory. The n = 2 case of the bulk-boundary correspondence is precisely the twistor correspondence where the boundary is playing the role of the twistor space and the bulk is a finite geometric version of compactified Minkowski space-time. For n ≥ 3 the bulk is identified with the finite geometric version of the Brody-Hughston quantum space-time. For special regions on both sides of the correspondence we associate certain collections of qubit observables. On the boundary side this association gives rise to the well-known GHW quantum net structure. In this picture the messages are complete sets of commuting observables associated to Lagrangian subspaces giving a partition of the boundary. Incomplete subsets of observables corresponding to subspaces of the Lagrangian ones are regarded as corrupted messages. Such a partition of the boundary is represented on the bulk side as a special collection of space-time points. For a particular message residing in the boundary, the set of possible errors is described by the fine details of the light-cone structure of its representative space-time point in the bulk. The geometric arrangement of representative space-time points, playing the role of the variety of codewords, encapsulates an algebraic algorithm for recovery from errors on the boundary side.

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Section: The Real Slice Of the Bulkmentioning

confidence: 99%

“…Since Lagrangian subspaces are (locally) represented by the arrangement (1|A) where A is symmetric, our real slice is an algebraic subvariety of the bulk of dimension n(n + 1)/2. The number of real points in the bulk is [64] n i=1 (1 + 2 i ), i.e. for n = 2, 3, 4 we have 15, 135, 2295 points.…”

Section: The Real Slice Of the Bulkmentioning

confidence: 99%

Finite geometric toy model of spacetime as an error correcting code

Lévay

Holweck

2019

Phys. Rev. D

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“…This, however, is quite far away from the correct value for line symplectic Grassmann codes, namely d min (W(n, 2)) = q 4n−5 − q 2n−3 , as we have determined in Section 3. We point out that in [7,Proposition 5], an upper bound on the minimum distance for Lagrangian-Grassmannian codes is given in terms of the dimension of the Lagrangian-Grassmannian variety, that is d min (W(n, n)) ≤ q n(n+1)/2 .…”

Section: Further Bounds On the Minimum Distancementioning

confidence: 99%

“…We point out that the code W(n, n) where k = n, corresponding to the so called dual polar space, has already been introduced under the name of Lagrangian-Grassmannian code of rank n in [7], where some bounds on the parameters have been obtained.…”

Section: Introductionmentioning

confidence: 99%

Minimum distance of symplectic Grassmann codes

Cardinali

Giuzzi

2016

Linear Algebra and its Applications

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In this paper we introduce symplectic Grassmann codes, in analogy to ordinary Grassmann codes and orthogonal Grassmann codes, as projective codes defined by symplectic Grassmannians. Lagrangian-Grassmannian codes are a special class of symplectic Grassmann codes. We describe all the parameters of line symplectic Grassmann codes and we provide the full weight enumerator for the Lagrangian-Grassmannian codes of rank 2 and 3

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“…In Section 4, computations will be handled using Maple and Macaulay2 to get the equations of the ideal of the Lagrangian Grassmannian LGr(N, 2N ) for N = 3 and N = 4. The sources of the codes are available at http://www.emis.de/journals/SIGMA/2014/041/codes.zip which contains two files: one is a Maple file to compute all equations defining the ideal of LGr (4,8), the other is a Macaulay2 script to compute the ideal of the projection of LGr (4,8) by elimination theory based on the equations stemming from the previous code. In what follows we shall only be concerned with W(2N − 1, 2); this space features |PG(2N − 1, 2)| = 2 2N − 1 = 4 N − 1 points and the number of its generators amounts to (2 + 1)(2 2 + 1) · · · (2 N + 1).…”

Section: Introductionmentioning

confidence: 99%

A Notable Relation between N-Qubit and 2^N-1-Qubit Pauli Groups via Binary LGr(N,2N)

Holweck¹

2014

SIGMA

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Abstract. Employing the fact that the geometry of the N -qubit (N ≥ 2) Pauli group is embodied in the structure of the symplectic polar space W(2N − 1, 2) and using properties of the Lagrangian Grassmannian LGr(N, 2N ) defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the N -qubit Pauli group and a certain subset of elements of the 2 N −1 -qubit Pauli group. In order to reveal finer traits of this correspondence, the cases N = 3 (also addressed recently by Lévay, Planat and Saniga [J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages]) and N = 4 are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space PG(2 N − 1, 2) of the 2 N −1 -qubit Pauli group in terms of G-orbits, where G ≡ SL(2, 2)×SL(2, 2)×· · ·×SL(2, 2) S N , to decompose π(LGr (N, 2N )) into non-equivalent orbits. This leads to a partition of LGr(N, 2N ) into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.

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On Lagrangian–Grassmannian codes

Cited by 19 publications

References 16 publications

Finite geometric toy model of spacetime as an error correcting code

Finite geometric toy model of spacetime as an error correcting code

Minimum distance of symplectic Grassmann codes

A Notable Relation between N-Qubit and 2^N-1-Qubit Pauli Groups via Binary LGr(N,2N)

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On Lagrangian–Grassmannian codes

Cited by 19 publications

References 16 publications

Finite geometric toy model of spacetime as an error correcting code

Finite geometric toy model of spacetime as an error correcting code

Minimum distance of symplectic Grassmann codes

A Notable Relation between N-Qubit and 2N-1-Qubit Pauli Groups via Binary LGr(N,2N)

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A Notable Relation between N-Qubit and 2^N-1-Qubit Pauli Groups via Binary LGr(N,2N)